KC Sinha: Exercise 5.1- Mathematics Solution Class 12 Chapter 5 आव्यूह

[mathjax]

Question 2
आव्यूह को संभव कांटियाँ क्या हैं यदि इसमे
(i) 5 अवयव हैं
(ii) 8  अवयव हैं
(iii) 24 अवयव हैं
(iv) 13 अवयव हैं
(v) 18 अवयव हैं
[What are the possible orders a matrix can have if it has]:
(i) 5 elements
(ii) 8 elements
(iii) 24 elements
(iv) 13 elements
(v) 18 elements

Sol :
(i) 1×5 , 5×1
(ii) 1×8 , 8×1 , 2×4 , 4×2
(iii)1×24 , 24×1 , 2×12 , 12×2 , 3×8 , 8×3 , 4×6 , 6×4
(iv) 1×13 , 13×1
(v) 1×18 , 18×1 , 2×9 , 9×2 , 3×6 , 6×3

Question 3
(i) एक 2×3 आव्यूह बनाएँ जिसका अवयव $a_{ij}=i+2j$ द्वारा प्रदत्त है ।
[Construct a 2×3 matrix whose elements are given by $a_{ij}=i+2j$]

Sol :

Let A=$\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\end{bmatrix}$

a_ij=i+2j

a₁₁=1+2(1)=1+2=3
a₁₂=1+2(2)=1+4=5
a₁₃=1+2(3)=1+6=7
a₂₁=2+2(1)=2+2=4
a₂₂=2+2(2)=2+4=6
a₂₃=2+2(3)=2+6=8

∴A=$\begin{bmatrix}3&5&7\\4&6&8\end{bmatrix}$

(ii) एक 3×2 आव्यूह की रचना करें जिसके अवयव $a_{ij}=\dfrac{1}{2}|i-3j|$ से प्रदत्त है ।
[Construct a 3×2 matrix whose elements are given by $a_{ij}=\dfrac{1}{2}|i-3j|$]

Sol :
Let A=$\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\a_{31}&a_{32}\end{bmatrix}$

$a_{ij}=\dfrac{1}{2}|i-3j|$

$a_{11}=\dfrac{1}{2}|1-3(1)|=\dfrac{1}{2}|-2|$ $=\dfrac{1}{2}\times 2=1$

$a_{12}=\dfrac{1}{2}|1-3(2)|=\dfrac{1}{2}|-5|$ $=\dfrac{1}{2}\times 5=\dfrac{5}{2}$

$a_{21}=\dfrac{1}{2}|2-3(1)|=\dfrac{1}{2}|-1|$ $=\dfrac{1}{2}\times 1=\dfrac{1}{2}$

$a_{22}=\dfrac{1}{2}|2-3(2)|=\dfrac{1}{2}|-4|$ $=\dfrac{1}{2}\times 4=2$

$a_{31}=\dfrac{1}{2}|3-3(1)|=\dfrac{1}{2}|0|=0$

$a_{32}=\dfrac{1}{2}|3-3(2)|=\dfrac{1}{2}|-3|$ $=\dfrac{1}{2}\times 3=\dfrac{3}{2}$

∴A=$\begin{bmatrix}1&\frac{5}{2}\\\frac{1}{4}&2\\0&\frac{3}{2}\end{bmatrix}$

Question 5

 एक 2×2 आव्यूह की रचना करें जिसके अवयव निम्नलिखित है :

[Construct a 2×2 matrix whose elements are]

(i)$a_{i j}=\frac{(i+j)^{2}}{2}$

Sol :

Let A $=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21}, & a_{22}\end{array}\right]$

$a_{i j}=\frac{(i+j)^{2}}{2}$

$a_{11}=\dfrac{(1+1)^{2}}{2}=\dfrac{4}{12}=2$

$a_{12}=\frac{(1+2)^{2}}{2}=\frac{9}{2}$

$a_{21}=\frac{(2+1)^{2}}{2}=\frac{9}{2}$

$a_{22}=\frac{(2+2)^{2}}{2}=\frac{16}{2}=8$

$A=\left[\begin{array}{ll}2 & \frac{5}{2} \\ \frac{9}{2} & 8\end{array}\right]$

(ii)$a_{i j}=\frac{(i+2 j)^{2}}{2}$

Sol :

Question 6

एक 3×4 आव्यूह $A=[A_{iij}]$ की रचना करें जिसके अवयव निम्न प्रकार से प्रदत्त हैं।

[Constuct a 3×4 matrix $A=[A_{iij}]$ whose elements are given by]

(i) $a_{i j}=i-j$

(ii) $a_{i j}=i+j$

(iii) $a_{i j}=i . j$

(iv) $a_{i j}=\frac{i}{j}$

(v) $a_{i j}=2 i-j$

(vi) $a_{i j}=\frac{1}{2}|-3 i+j|$

Sol :

आव्यूह $\left[\begin{array}{rrr}1 & 0 & 5 \\ 2 & -3 & 4\end{array}\right]$ में (In the matrix) $\left[\begin{array}{lrl}1 & 0 & 5 \\ 2 & -3 & 4\end{array}\right]$

(i) पोक्तियों की संख्या ___ है। 

[Number of rows is ___]

(ii) स्तम्भों की संख्या ___ है।

[Number of column is ___]

(iii) आव्यूह की कोटि ___ है।

[Order of the matrix is ___]

(iv) प्रविष्टियों की संख्या ___ है।

[Number of entries is ___ ]

Sol :

माना $\mathrm{A}=\left[\begin{array}{rrr}2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & -1\end{array}\right]$ , तो निम्नलिखित में कौन सत्य है।

Let $\mathrm{A}=\left[\begin{array}{rrr}2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & -1\end{array}\right]$ then which of the following is true

(i) A एक विकर्ण आव्यूह है लेकिन अदिश आव्यूह नहीं है।

A is a diagonal matrix but not a scalar matrix 

Sol :

[True]

(ii) A एक अदिश आव्यूह है लेकिन विर्कण आव्यूह नहीं है।

A is a scalar matrix but not a diagonal matrix 

Sol :

[False]

(iii) A विकर्ण आव्यूह तथा अदिश आव्यूह दोनो है।

A is a diagonal matrix as well as scalar matrix

A यदि पंक्ति आव्यूह तथा स्तम्भ आव्यूह दोनों है तो A का कोटि है__

[ If A is a row matrix as well as a column matrix , then order of A is ___ ]
Sol :
1×1 
or

[a₁₁]

Question 10

( माना ) Let $\mathrm{A}=\left[a_{i j}\right]=\left[\begin{array}{rrr}1 & -2 & 5 \\ 3 & 4 & -6 \\ 9 & 15 & 13\end{array}\right]$ ( तथा )and $\mathrm{B}=\left[b_{i j}\right]=\left[\begin{array}{cccc}2 & 5 & 19 & -7 \\ 35 & -2 & \frac{5}{2} & 12 \\ \sqrt{3} & 1 & -5 & 17\end{array}\right]$

आव्यूह A का कोटि है।

[Order of the matrix A is]
Sol :
3×3

(ii) a₂₃=
Sol : -6

(iii) a₃₁=
Sol : -9

(iv)

[Order of the matrix B is ]

Sol : 3×4

(v) B में अवयवों की संख्या ___ है।

[The number of elements in B is]
Sol : 12

(vi)

अवयवों b₁₃,b₂₁,b₃₃,b₂₄,b₂₃  को लिखें।

[Write the elements b₁₃,b₂₁,b₃₃,b₂₄,b₂₃ ]

Sol :
b₁₃=19
b₂₁=35
b₃₃=-5
b₂₄=12
b₂₃=$\dfrac{5}{2}$

Question 11

सुधा तथा उसकी दो सहेलियाँ सयिदा तथा सिमरन के पास पुस्तिकाओं तथा कलमों की संख्या के बारे में निम्नलिखित सूचनाओं पर विचार करें।

[Consider the following information regarding number of note books and pens possessed by Sudha and her two friends Syeeda and Simran]

	पुस्तिकाओं की संख्या (Number of note nooks)	कलमों की संख्या (Number of pens)
सुधा (Sudha)	15	6
सयिदा (Syeeda)	10	2
सिमरन(Simran)	13	5

ऊपर दिए गए सूचनाओं को 3×2 आव्यूह के रूप में लिखें। दूसरी पंक्ति और पहले स्तम्भ की प्रविष्टि क्या प्रकट करती है ?

[Represent the above informations in the form of a 3×2 matrix. What does the entry in the second row and first column represent ?]

Sol :

Question 12
(i) यदि (if) $\left[\begin{array}{cc}x+y & 2 \\ 1 & x-y\end{array}\right]=\left[\begin{array}{ll}3 & 2 \\ 1 & 7\end{array}\right]$ , तो (then) x=__ , y=__
Sol :
$\begin{aligned}
x+y &=3..(i) \\
x-y &=7..(ii) \\
\hline 2 x &=10\\x=5
\end{aligned}$

Putting the value of x in equation (i)
⇒x+y=3
⇒5+y=3
⇒y=3-5=-2

(ii) यदि (if) $\left[\begin{array}{cc}x-y & 2 x-x_{1} \\ 2 x-y & 3 x+y_{1}\end{array}\right]=\left[\begin{array}{cc}-1 & 5 \\ 0 & 13\end{array}\right]$ ,
तथा P और Q के नियामक क्रमश
(x,y) तथा (x₁y₁) है तो PQ=__
[and co-ordinates of points P and Q be (x,y) and (x₁y₁) respectively] then PQ=__
Sol :
$\begin{aligned} x-y &=-1 ..(i)\\ 2 x-y &=0 ..(ii)\\ \hline x&=1 \end{aligned}$

⇒1-y=-1
⇒y=2

$\begin{aligned} 2 x-x_{1} &=5..(iii) \\ 3 x+y_{1} &=13...(iv) \end{aligned}$

From equation (iii)
⇒$2 x-x_{1}=5$
⇒$2 (1)-x_{1}=5$
⇒x₁=-3

From equation (iv)

$3 x+y_{1}=13$

$3(1)+y_{1}=13$
$y_{1}=10$

Diagram

$PQ=\sqrt{(-3-1)^2+(10-2)^2}$
=$\sqrt{-4^2+8^2}$
=$\sqrt{16+64}=\sqrt{80}$
=$4\sqrt{5}$
Question 13
(i) यदि (if) $\left[\begin{array}{cc}x-y & 2 x+z \\ 2 x-y & 3 z+\omega\end{array}\right]=\left[\begin{array}{rr}-1 & 5 \\ 0 & 13\end{array}\right]$  , (find) x, y,z,ω , निकाले
Sol :
$\begin{aligned} x-y &=-1..(i) \\2 x+y &=0..(ii) \\\hline x &=1 \end{aligned}$

Putting x=1 in equation (1)
⇒1-y=-1
⇒y=2

2x+z=5..(iii)

3z+ω=13...(iv)

Putting x=1 in equation (iii)
⇒2x+z=5
⇒2(1)+z=5
⇒z=3

putting z=3 in equation (iv)
⇒3(3)+ω=13
⇒ω=4

(ii) यदि (If) $\left[\begin{array}{cc}x-y & z \\ 2 x-y & \omega\end{array}\right]=\left[\begin{array}{rr}-1 & 4 \\ 0 & 5\end{array}\right]$, (find) $x, y, z, \omega$ निकालें।

Sol :

(iii) यदि (If) $\left[\begin{array}{cc}x & 3 x-y \\ 2 x+z & 3 y-\omega\end{array}\right]=\left[\begin{array}{ll}3 & 2 \\ 4 & 7\end{array}\right]$, (find) $x, y, z, \omega$ निकाले।

Sol :

Question 14
निम्नलिखित समिकरणों x,y,z निकाले ।
[Find x,y,z from the following equations]
(i) $\left[\begin{array}{cc}x+y & 2 \\ 5+z & x y\end{array}\right]=\left[\begin{array}{cc}6 & 2 \\ 5 & 8\end{array}\right]$
Sol :
x+y=6⇒y=6-x

⇒xy=8
⇒x(6-x)=8
⇒6x-x²=8
⇒x²-6x+8=0
⇒x²-4x-2x+8=0
⇒x(x-4)-2(x-4)=0
⇒(x-4)(x-2)=0

⇒x-4=0 , x-2=0
⇒x=4 , x=2

If x=4 , then y=6-4=2

If x=2 , then y=6-2=4

5+z=5
z=0

x+4 , y=2 , z=0
x=2 , y=4 , z=0

(ii) $\left[\begin{array}{ll}4 & 3 \\ x & 5\end{array}\right]=\left[\begin{array}{ll}y & z \\ 1 & 5\end{array}\right]$
Sol :

y=4

z=3

x=1

(iii) $\left[\begin{array}{c}x+y+z \\ x+z \\ y+z\end{array}\right]=\left[\begin{array}{l}9 \\ 5 \\ 7\end{array}\right]$
Sol :
⇒x+y+z=9..(i)
⇒x+z=5..(ii)
⇒y+z=7..(iii)

⇒x+y+z=9
⇒5+y=9
⇒y=4

Putting y=4 in equation (iii)
⇒4+z=7
⇒z=3

Putting z=3 in equation (ii)
⇒x+3=5
⇒x=2

Question 15
a, b,c,d निकाले यदि (Find a,b,c,d if) $\left[\begin{array}{cc}2 a+b & a-2 b \\ 5 c-d & 4 c+3 d\end{array}\right]=\left[\begin{array}{cc}4 & -3 \\ 11 & 24\end{array}\right]$
Sol :
2a+b=4..(i)
a-2b=-3..(ii)
5c-d=11..(iii)
4c+3d=24...(iv)

Question 16
माना कि (let) $A=\left[\begin{array}{ll}2 & 4 \\ 3 & 2\end{array}\right]$,$ B=\left[\begin{array}{rr}1 & 3 \\ -2 & 5\end{array}\right]$, $C=\left[\begin{array}{rr}-2 & 5 \\ 3 & 4\end{array}\right]$ निकाले (find)
(i) A+B
Sol :
$A+B=\left[\begin{array}{cc}2 & 4 \\ 3 & 2\end{array}\right]+\left[\begin{array}{cc}1 & 3 \\ -2 & 5\end{array}\right]$

$=\left[\begin{array}{ll}3 & 7 \\ 1 & 7\end{array}\right]$

(ii) A-B
Sol :
$A-B=\left[\begin{array}{ll}2 & 4 \\ 3 & 2\end{array}\right]-\left[\begin{array}{cc}1 & 3 \\ -2 & 5\end{array}\right]$

$=\left[\begin{array}{cc}1 & 1 \\ 5 & -3\end{array}\right]$

(iii) 3A-C
Sol :
$3 A-C=3\left[\begin{array}{cc}2 & 4 \\ 3 & 2\end{array}\right]-\left[\begin{array}{rr}-2 & 5 \\ 3 & 4\end{array}\right]$

$=\left[\begin{array}{cc}6 & 12 \\ 9 & 6\end{array}\right]-\left[\begin{array}{cc}-2 & 5 \\ 3 & 4\end{array}\right]$

$=\left[\begin{array}{ll}8 & 7 \\ 6 & 2\end{array}\right]$

Question 18
निमनलिखित का परिकलन करे (Compute the following)
(i) $\left[\begin{array}{rrr}0 & 1 & 5 \\ -3 & 2 & 1\end{array}\right]+\left[\begin{array}{rrr}6 & 2 & -3 \\ -1 & 4 & 2\end{array}\right]$
Sol :
$\left[\begin{array}{ccc}0 & 1 & 5 \\ -3 & 2 & 1\end{array}\right]+\left[\begin{array}{ccc}6 & 2 & -3 \\ -1 & 4 & 2\end{array}\right]$

(ii) $\left[\begin{array}{rr}2 & -1 \\ 3 & 5\end{array}\right]+\left[\begin{array}{rr}4 & 3 \\ 1 & -2\end{array}\right]$
Sol :

(iii) $\left[\begin{array}{rr}a & b \\ -b & a\end{array}\right]+\left[\begin{array}{ll}a & b \\ b & a\end{array}\right]$
Sol :
$=\left[\begin{array}{cc}2 a&2b \\ 0 & 2 a\end{array}\right]$

(iv) $\left[\begin{array}{cc}\cos ^{2} x & \sin ^{2} x \\ \sin ^{2} x & \cos ^{2} x\end{array}\right]+\left[\begin{array}{cc}\sin ^{2} x & \cos ^{2} x \\ \cos ^{2} x & \sin ^{2} x\end{array}\right]$
Sol :
$=\left[\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right]$

(v) $\left[\begin{array}{cc}a^{2}+b^{2} & b^{2}+c^{2} \\ a^{2}+c^{2} & a^{2}+b^{2}\end{array}\right]+\left[\begin{array}{cc}2 a b & 2 b c \\ -2 a c & -2 a b\end{array}\right]$
Sol :
$=\left[\begin{array}{ll}a^{2}+b^{2}+2 a b & b^{2}+c^{2}+2 b c \\ a^{2}+c^{2}-2 a c & a^{2}+b^{2}-2 ab\end{array}\right]$

$=\left[\begin{array}{ll}(a+b)^{2} & (b+c)^{2} \\ (a-c)^{2} & (a-b)^{2}\end{array}\right]$

Question 20
यदि (if) $A=\left[\begin{array}{cc}2 & -1 \\ 4 & 2\end{array}\right]$,$ B=\left[\begin{array}{rr}4 & 3 \\ -2 & 1\end{array}\right]$,$ C=\left[\begin{array}{rr}-2 & -3 \\ -1 & 2\end{array}\right]$
निम्नलिखित का परिकलन करे (compute the following)
(i) A+(B+C)
Sol :
$=\left[\begin{array}{cc}2 & -1 \\ 4 & 2\end{array}\right]+\left(\left[\begin{array}{cc}4 & 3 \\ -2 & 1\end{array}\right]+\left[\begin{array}{cc}-2 & -3 \\ -1 & 2\end{array}\right]\right)$

$=\left[\begin{array}{cc}2 & -1 \\ 4 & 2\end{array}\right]+\left[\begin{array}{cc}2 & 0 \\ -3 & 3\end{array}\right]$

$=\left[\begin{array}{cc}4 & -1 \\ 1 & 5\end{array}\right]$

(ii) (A+B)+C
Sol :

(iii)-2A+(B+C)

(iv)A+(2B-C)
Sol :
$=\left[\begin{array}{cc}2 & -1 \\ 4 & 2\end{array}\right]+\left(2\left[\begin{array}{cc}4 & 3 \\ -2 & 1\end{array}\right]-\left[\begin{array}{cc}-2 & -3 \\ -1 & 2\end{array}\right]\right)$

$=\left[\begin{array}{cc}2 & -1 \\ 4 & 2\end{array}\right]+\left(\left[\begin{array}{cc}8 & 6 \\ -4 & 2\end{array}\right]-\left[\begin{array}{cc}-2 & -3 \\ -1 & 2\end{array}\right]\right)$

$=\left[\begin{array}{cc}2 & -1 \\ 4 & 2\end{array}\right]+\left[\begin{array}{cc}10 & 9 \\ -3 & 0\end{array}\right]$

$=\left[\begin{array}{cc}12 & 8 \\ 1 & 2\end{array}\right]$

Question 21
यदि (if) $A=\left[\begin{array}{rrr}1 & 2 & 3 \\ -1 & 0 & 2 \\ 1 & -3 & -1\end{array}\right], B=\left[\begin{array}{rrr}4 & 5 & 6 \\ -1 & 0 & 1 \\ 2 & 1 & 2\end{array}\right]$ तथा (and) $C=\left[\begin{array}{rrr}-1 & -2 & 1 \\ -1 & 2 & 3 \\ -1 & -2 & 2\end{array}\right]$

A+(B+C)=(A+B)+C को सत्यापित करे ।
[verify that A+(B+C)=(A+B)+C]
Sol :
L.H.S
A+(B+C)

$=\left[\begin{array}{ccc}1 & 2 & 3 \\ -1 & 0 & 2 \\ 1 & -3 & -1\end{array}\right]+\left(\left[\begin{array}{ccc}4 & 5 & 6 \\ -1 & 0 & 1 \\ 2 & 1 & 2\end{array}\right]+\left[\begin{array}{ccc}-1 & -2 & 1 \\ -1 & 2 & 3 \\ -1 & -2 & 2\end{array}\right]\right)$

$=\left[\begin{array}{ccc}1 & 2 & 3 \\ -1 & 0 & 2 \\ 1 & -3 & -1\end{array}\right]+\left[\begin{array}{ccc}3 & 3 & 7 \\ -2 & 2 & 4 \\ 1 & -1 & 4\end{array}\right]$

$=\left[\begin{array}{ccc}4 & 5 & 10 \\ -3 & 2 & 6 \\ 2 & -4 & 3\end{array}\right]$

R.H.S
(A+B)+C

$=\left(\left[\begin{array}{ccc}1 & 2 & 3 \\ -1 & 0 & 2 \\ 1 & -3 & -1\end{array}\right]+\left[\begin{array}{ccc}4 & 5 & 6 \\ -1 & 0 & 1 \\ 2 & 1 & 2\end{array}\right]\right)+\left[\begin{array}{ccc}-1 & -2 & 1 \\ -1 & 2 & 3 \\ -1 & -2 & 2\end{array}\right]$

$=\left[\begin{array}{ccc}5 & 7 & 9 \\ -2 & 0 & 3 \\ 3 & -2 & 1\end{array}\right]+\left[\begin{array}{ccc}-1 & -2 & 1 \\ -1 & 2 & 3 \\ -1 & -2 & 2\end{array}\right]$

$=\left[\begin{array}{ccc}4 & 5 & 10 \\ -3 & 2 & 6 \\ 2 & -4 & 3\end{array}\right]$

L.H.S=R.H.S

Question 22
निकाले (Evaluate) $\begin{bmatrix}\sin ^{2} \theta & 1\\\cot ^{2} \theta & 0\end{bmatrix}+\left[\begin{array}{cc}\cos ^{2} \theta & 0 \\ -\operatorname{cosec}^{2} \theta & 1\end{array}\right]+\left[\begin{array}{cc}0 & -1 \\ -1 & 0\end{array}\right]$
Sol :
$=\left[\begin{array}{ll}\sin ^{2} \theta+cos^{2} \theta+0 & 1+0-1 \\ \cot ^{2} \theta-cosec ^{2} \theta-1 & 0+1+0\end{array}\right]$

$=\left[\begin{array}{cc}1 & 0 \\ -1-1 & 1\end{array}\right]$

$=\left[\begin{array}{cc}1 & 0 \\ -2 & 1\end{array}\right]$

Question 23
(i) निम्नलिखित समीकरण से x और y निकाले
[From the following equations , find the values of x and y]
$2\left[\begin{array}{cc}x & 5 \\ 7 & y-3\end{array}\right]+\left[\begin{array}{cc}3 & 4 \\ 1 & 2\end{array}\right]=\left[\begin{array}{cc}7 & 14 \\ 15 & 14\end{array}\right]$
Sol :
$\left[\begin{array}{cc}2 x & 10 \\ 14 & 2 y-6\end{array}\right]+\left[\begin{array}{cc}3 & 4 \\ 1 & 2\end{array}\right]=\left[\begin{array}{cc}7 & 14 \\ 15 & 14\end{array}\right]$

$\left[\begin{array}{cc}2 x+3 & 14 \\ 15 & 2 y-4\end{array}\right]=\left[\begin{array}{cc}7 & 14 \\ 15 & 14\end{array}\right]$

⇒2x+3=7
⇒2x=4
⇒x=2

⇒2y-4=14
⇒2y=18
⇒y=9

(ii) निम्नलिखित समीकरण को संतुष्ट करने वाले x और y का मान निकालें। 

[Find the values of x and y satisfying the following equation]

$2\left[\begin{array}{cc}x & 5 \\ 7 & y-3\end{array}\right]+\left[\begin{array}{lr}3 & -4 \\ 1 & 2\end{array}\right]=\left[\begin{array}{cc}7 & 6 \\ 15 & 14\end{array}\right]$

Sol :

Question 24

निम्नलिखित समीकरण को संतुष्ट करने वले x, y, z तथा t का मान निकालें । 

[Find x, y, z and t satisfying the following.equation] :

$2\left[\begin{array}{ll}x & z \\ y & t\end{array}\right]+3\left[\begin{array}{lr}1 & -1 \\ 0 & 2\end{array}\right]=3\left[\begin{array}{ll}3 & 5 \\ 4 & 6\end{array}\right]$

Sol :

Question 25
यदि A=विकर्ण [1 2  3]
B=विकर्ण [0 2  5]
C=विकर्ण [3 -2  5]

If A=diag [1 2  3]
B=diag [0 2  5]
C=diag [3 -2  5]

(i)4A-3B
Sol :
=diag[4  12  8]-diag[0  6  15]
=diag[4  6  -7]

(ii) A+B-2C
Sol :

Question 26
आव्यूह X निकालें यदि (Find matrix X , if ) $X+\left[\begin{array}{ll}2 & 5 \\ 3 & 2\end{array}\right]=\left[\begin{array}{rr}4 & 0 \\ -7 & 6\end{array}\right]$
Sol :
$x=\left[\begin{array}{cc}4 & 0 \\ -7 & 6\end{array}\right]-\left[\begin{array}{ll}2 & 5 \\ 3 & 2\end{array}\right]$

$x=\left[\begin{array}{rr}2 & -5 \\ -10 & 4\end{array}\right]$

Question 27
आव्यूह X निकालें ताकि (Find a matrix X such that) 2A+B+X=0
जहाँ (where) $A=\left[\begin{array}{rr}-1 & 2 \\ 3 & 4\end{array}\right]$  तथा (and) $B=\left[\begin{array}{rr}3 & -2 \\ 1 & 5\end{array}\right]$
Sol :
2A+B+X=0

$2\left[\begin{array}{cc}-1 & 2 \\ 3 & 4\end{array}\right]+\left[\begin{array}{cc}3 & -2 \\ 1 & 5\end{array}\right]+X=0$

$\left[\begin{array}{cc}-2 & 4 \\ 6 & 8\end{array}\right]+\left[\begin{array}{cc}3 & -2 \\ 1 & 5\end{array}\right]+X=0$

$\left[\begin{array}{ll}1 & 2 \\ 7 & 13\end{array}\right]+X=0$

$X=\left[\begin{array}{cc}-1 & -2 \\ -7 & -13\end{array}\right]$

Question 28
(i) आव्यूह X निकालें ताकि ( Find a matrix X such that )

A+2B+X=0 ( जहाँ ) where
$A=\left[\begin{array}{rr}2 & -1 \\ 3 & 5\end{array}\right] ; B=\left[\begin{array}{rr}-1 & 1 \\ 0 & 2\end{array}\right]$

(ii)  3×2 कोटि का एक आव्यूह X निकालें ताकि ( Find a matrix X of order 3×2 such that) 2A+3X=5 , जहाँ (where) $A=\left[\begin{array}{rr}8 & 0 \\ 4 & -2 \\ 3 & 6\end{array}\right]$ तथा (and) $B=\left[\begin{array}{rr}2 & -2 \\ 4 & 2 \\ -5 & 1\end{array}\right]$

Sol :

Question 29
आव्यूह X तथा Y निकालें यदि (Find matrices X and Y , if) $x+y=\left[\begin{array}{ll}5 & 2 \\ 0 & 9\end{array}\right]$ तथा (and) $\mathrm{X}-\mathrm{Y}=\left[\begin{array}{ll}3 & \phantom{-}6 \\ 0 & -1\end{array}\right]$
Sol :
$\begin{aligned} X+Y &=\left[\begin{array}{ll}5 & 2 \\ 0 & 3\end{array}\right]..(i) \\ X-Y &=\left[\begin{array}{ll}3 & \phantom{-}6 \\ 0 & -1\end{array}\right] ..(ii)\\\hline 2 x &=\left[\begin{array}{ll}8 & 8 \\ 0 & 8\end{array}\right] \end{aligned}$

$X=\left[\begin{array}{cc}4 & 4 \\ 0 & 4\end{array}\right]$

Putting the value of X in equation (i)

$\left[\begin{array}{ll}4 & 4 \\ 0 & 4\end{array}\right]+Y=\left[\begin{array}{ll}5 & 2 \\ 0 & 9\end{array}\right]$

$y=\left[\begin{array}{ll}5 & 2 \\ 0 & 9\end{array}\right]-\left[\begin{array}{cc}4 & 4 \\ 0 & 4\end{array}\right]$

$y=\left[\begin{array}{cc}1 & -2 \\ 0 & 5\end{array}\right]$

Question 30
दिखाएँ कि (Show that)
$\cos \theta\left[\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right]+\sin \theta\left[\begin{array}{cr}\sin \theta & -\cos \theta \\ \cos \theta & \sin \theta\end{array}\right]=I$
Sol :
L.H.S
$\cos \theta\left[\begin{array}{rr}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta\end{array}\right]+\sin \theta\left[\begin{array}{rr}\sin \theta & -\cos \theta \\ \cos \theta & \sin \theta\end{array}\right]$

$=\left[\begin{array}{cc}\cos ^{2} \theta & \cos \theta \sin \theta \\ -\cos \theta \sin \theta & \cos ^{2} \theta\end{array}\right]+\left[\begin{array}{cc}\sin ^{2} \theta & -\cos \theta\sin \theta \\ \cos \theta \sin \theta & \sin ^2 \theta\end{array}\right]$

$=\left[\begin{array}{ccc}\cos ^{2} \theta+sin^2 \theta & 0 \\ 0 & \cos ^{2} \theta+\sin ^{2} \theta\end{array}\right]$

$=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]=I$

Question 31
यदि (if) $2 \mathrm{X}+3 \mathrm{Y}=\left[\begin{array}{ll}2 & 3 \\ 4 & 0\end{array}\right]$ तथा (and) $3 X+2 Y=\left[\begin{array}{rr}-2 & 2 \\ 1 & -5\end{array}\right] X$ तथा Y निकाले । [Find X and Y]
Sol :
$\begin{aligned} 4 x+6 y &=\left[\begin{array}{rr}4 & 6 \\ 8 & 0\end{array}\right] \\ 9 x+4 y &=\left[\begin{array}{cc}-6 & 6 \\ 3 & -15\end{array}\right] \\ \hline-5 x &=\left[\begin{array}{cc}10 & 0 \\ 5 & 15\end{array}\right] \end{aligned}$

$x=\left[\begin{array}{cc}-2 & 0 \\ -1 & 3\end{array}\right]$

Putting the value of x in equation (i)

$2 x+3 y=\left[\begin{array}{ll}2 & 3 \\ 4 & 0\end{array}\right]$

$2\left[\begin{array}{cc}-2 & 0 \\ -1 & -3\end{array}\right]+3 y=\left[\begin{array}{cc}2 & 3 \\ 4 & 0\end{array}\right]$

$\left[\begin{array}{cc}-4 & 0 \\ -2 & -6\end{array}\right]+3 x=\left[\begin{array}{ll}2 & 3 \\ 4 & 0\end{array}\right]$

$3 x=\left[\begin{array}{cc}2 & 3 \\ 4 & 0\end{array}\right]-\left[\begin{array}{cc}-4 & 0 \\ -2 & -6\end{array}\right]$

$3 y=\left[\begin{array}{ll}6 & 3 \\ 6 & 6\end{array}\right]$

$Y=\left[\begin{array}{ll}2 & 1 \\ 2 & 2\end{array}\right]$

Question 32

दिया है (Given) $A=\begin{bmatrix}1&2&-3\\5&0&2\\1&-1&1\end{bmatrix}$ तथा (and) $B=\begin{bmatrix}3&-1&2\\4&2&5\\2&0&3\end{bmatrix}$ , आव्यूह C निकाले ताकि A+C=B (Find the matrix C such that) A+C=B

Sol :

Question 33

यदि (If) $A=\begin{bmatrix}2&3&4\\-3&0&2\end{bmatrix}$ , $B=\begin{bmatrix}3&-4&-5\\1&2&1\end{bmatrix}$ तथा (and) $C=\begin{bmatrix}5&-1&2\\7&0&3\end{bmatrix}$ , आव्यूह X निकाले ताकि (Find the matrix X such that) 2A+3B=X+C

Sol 

Question 34

यदि (If) $A=\begin{bmatrix}2&3&-4\\1&0&6\\-2&1&5\end{bmatrix}$ 2A-3B निकालें (find 2A-3B)

Sol :

Question 35
यदि (If) $A=\begin{bmatrix}1&2&3\\-1&0&2\\-2&-3&1\end{bmatrix}$ ,$B=\begin{bmatrix}4&5&1\\-1&0&3\\2&1&2\end{bmatrix}$, $C=\begin{bmatrix}-1&-2&1\\-1&2&3\\-1&-2&2\end{bmatrix}$

A-2B+3C निकाले (Find A-2B+3C)

(Also verify that) (A+B)+C=A+(B+C) को भी सत्यापित करें।
Sol :

फातिमा की स्थान P तथा स्थान Q पर दो फैक्ट्रियाँ है। प्रत्येक फैक्ट्री में लड़कों तथा लड़कियों के लिए खेल के जूते, तीन भिन्न-भिन्न मूल्य वर्गो, क्रमशः 1 , 2 तथा 3 के बनते हैं। प्रत्येक फैक्ट्री मे बनने वाले जूतो की संख्या नीचे दिए आव्यूहों द्वारा निरूपित हैः

[Fatima has tw factories at places P and Q . Each factory produces sport shes for boys and girls in three different price categories labelled 1 , 2 and 3. The quantities produced by each factory are represented as matrices given below]

$A=\begin{array}{ccc}\text{Factory at P}\\(boys, girls)\\\left[\begin{array}{ll}95 & 60\\75&65 \\ 90 & 85\end{array}\right] \begin{array}{l}1 \\ 2 \\ 3\end{array} \end{array}$

$A=\begin{array}{ccc}\text{Factory at Q}\\(boys, girls)\\\left[\begin{array}{ll}90 & 50\\70&55 \\ 75 & 75\end{array}\right] \begin{array}{l}1 \\ 2 \\ 3\end{array} \end{array}$

ज्ञात करे (Find)

(i) प्रत्येक मूल्य वर्ग में लड़कों तथा लड़कियों के लिए बनने वाले खेल के जूतों की कुल संख्या।

[The total production of sport shes for boys and girls in each price category]

Sol :

(ii) प्रत्येक मूल्य वर्ग ंमें स्थान P तथा स्थान Q पर फैक्ट्री द्वारा लड़के तथा लड़कियो  के लिए खेल के जूतो की संंख्याओ का अन्तर
[The difference of sport shoes produced by factories at P and Q for boys and girls in each price category]
Sol :

$A-B=\left[\begin{array}{ll}5 & 10 \\ 5 & 10 \\ 15 & 10\end{array}\right] \begin{array}{l}1 \\ 2 \\ 3\end{array}$